# MHD Flow and Heat Transfer over Vertical Stretching Sheet with Heat Sink or Source Effect

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## Abstract

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## 1. Introduction

## 2. Problem Description

- $\sigma $ is the electrical conductivity,
- ${\mu}_{e}$ is the magnetic permeability,
- ${T}_{\infty}$ is the temperature of free stream,
- g is acceleration due to gravity,
- $\beta $ is the volumetric coefficient of thermal expansion,
- k is the thermal conductivity,
- $\upsilon \left(=\mu /\rho \right)$ is the kinematic viscosity, and
- ${T}_{w}={T}_{\infty}+bx$ is the temperature of the sheet.

## 3. Scheme Analysis

## 4. Stability Analysis

**Step 1:**To convert the governing Equations (2) and (3) of fluid flow in unsteady form, we have

**Step 2:**To introduce a new non-dimensional time variable $\tau =a.t$, and all other similarity variables are also a function of $\tau $, can be written as,

**Step 3:**By applying Equation (14) on Equations (12) and (13), we have

**Step 4:**To check the stability of steady flow solutions $f\left(\eta \right)={f}_{0}\left(\eta \right)\mathrm{and}\theta \left(\eta \right)={\theta}_{0}\left(\eta \right)$ will satisfy the basic model by introducing the following functions

**Step 5:**By putting Equation (18) into Equations (15) and (16) and keeping $\tau =0$, we have

**Step 6:**To relax one boundary condition into an initial condition, as suggested by Weidman et al. [36] and Harris et al. [34]. In this problem, we relaxed ${G}_{0}\left(\eta \right)\to 0,as\eta \to \infty $ into ${G}_{0}^{\prime}(\left(0\right)=1.$ We have to solve Equation (19) and (20) with boundary and relaxed initial condition in order to find the values of smallest eigenvalue γ.

^{1}continuous solution that is fourth order accurate uniformly in [a,b]. Mesh selection and error control are based on the residual of the continuous solution”. As we know, only the first solution is the stable and only the stable solution has physical meaning. In these regards, the various effect of different physical parameters on velocity and temperature profiles have been demonstrated for the first solution only. Finally, from Figure 1a,b, we draw some graphs in order to show the existence of multiple solutions for the opposing flow case.

## 5. Results and Discussion

_{i}= 1. It is perceived that, for assisting flow (λ > 0), the dimensionless velocity is the maximum at the superficial of the vertical stretching sheet and it increasingly reduces to the minimum value ${f}^{\prime}=1$ as it changes to gone after the superficial, while for opposing flow (λ < 0), the dimensionless velocity is the lowest at the surface of the vertical stretching sheet and gradually increases to the maximum value ${f}^{\prime}=1$, as it changes away from the superficial, this effect is mathematically obvious in Equation (10). It is additional observed that the velocity profile decreases with the Hartmann number for the assisting flow, whereas, it increases with the Hartmann number for the opposing flow. Consequently, the hydrodynamic boundary layer thickness also depends upon Ha. It is well known that the Hartmann number represents the proportion of electromagnetic force to the viscous force, thus, in the case of assisting flow, increasing the Hartmann number means that electromagnetic force was enhanced when compared to viscous force, which in turn Lorentz force augments, then opposes the flow, and then reduces the velocity profile. Nonetheless, in the situation of opposing flow, there is a reverse effect of the Hartmann number on dimensionless velocity. It should be pointed out that, in the case of assisting flow, (λ > 0) means the heating of the fluid, therefore the thermal buoyancy forces were enhanced. It can be interpreted on this fact that the highest value of dimensionless velocity is near the stretched surface; however, for opposing flow (λ > 0), which means that the fluid is consequently cooled; the thermal buoyancy forces decreases and then we realize the lowest value of dimensionless velocity near the stretched surface. It is worthwhile to note that the velocity profile increases with the mixed convection parameter λ for mutually case opposing and assisting flow due to an increasing of the thermal buoyancy forces. It can be seen that, for buoyancy opposed (λ < 0, opposing), the velocity profile will be significantly affected.

_{i}= 1, and Hartmann number Ha = 1. It can be seen that the velocity profile increases with stretching velocity ratio when stretching in the flow direction. Whereas, the velocity profile decreases with stretching velocity ratio when stretching in the opposite direction, this can be attributed to the significant enhancement in pressure on the sheet. Furthermore, it is remarked that the velocity profile augment with mixed convection parameter λ for both stretching in the flow direction and stretching in the opposite direction, as proven in Figure 1. The physical reason behind this is that, by augmenting the mixed convection parameter, the thermal buoyancy forces rise and help to push the flow in y direction, which in turn increases the velocity profile.

_{r}= H

_{a}= A = 1. It is clear that the temperature profile is greater at the stretching sheet surface and then exponentially lessens along the streamwise path up to the zero value for both assisting and opposing flow; this effect is proved and designated by the choice of boundary conditions and it is mathematically noticeable in Equation (10). It is worthwhile to note that the dimensionless temperature augment with both Biot number Bi and Heat generation/absorption coefficient δ for both case assisting and opposing flow; therefore, the thermal boundary layer thickness increases. It is well known that the Biot number signifies the proportion of heat convection to heat conduction; therefore increasing the Biot number leads to more heat will be released to the fluid flow, which in turn augments the temperature profile. Similarly, augmenting heat source leads to applying more heat to the fluid flow and it results in enhancing the dimensionless temperature.

_{r}= H

_{a}= A = 1. As expected, the temperature profile satisfies the boundary conditions, starting with a higher value at the surface of the sheet and then meaningfully declining to zero value when η increases. Furthermore, it is remarked that dimensionless temperature increases with Biot number, even in the company of heat sink, as shown in Figure 3 in the circumstance of heat basis, so it can be concluded that the temperature profile increase with biot number independent of Heat generation/absorption coefficient δ. It should be pointed out that, for positive values of Heat generation/absorption coefficient, δ acts as a heat source, but for a negative value of Heat generation/absorption coefficient, δ acts as sink source, this signifies that the dimensionless temperature reduced in case of a negative value of δ (heat sink) when compared to a positive value of δ (heat source), as presented in Figure 3. Similarly, the thermal boundary layer thickness decreases.

## 6. Conclusions

- It is perceived that, for assisting flow (λ > 0), the dimensionless velocity is the maximum at the superficial of the vertical stretching sheet and it gradually lessens to the minimum value ${f}^{\prime}=1$, as it transfers to gone after the superficial, while for opposing flow (λ < 0), the dimensionless velocity is the lowest at the superficial of the vertical stretching sheet and gradually increases to the maximum value ${f}^{\prime}=1$.
- The velocity profile augments with mixed convection parameter λ for both stretching in the flow direction and stretching in the opposite direction.
- The temperature profile is greater at the stretching sheet superficial then exponentially lessens along the streamwise path up to the zero value for both assisting and opposing flow.
- Skin friction increase with a mixed convection parameter and Hartmann number Ha, though it declines by stretching speed ratio for together opposing and assisting flow.
- The effect of mixed convection parameter on the heat transfer rate is slightly perceptible. The Nusselt number at the sheet surface augments, because the Hartmann number, Hartmann number Ha, stretching velocity ratio A, and mixed convection parameter λ increase. Though, it declines according to heat generation/absorption coefficient δ.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The existence of multiple solutions for the opposing flow case different parameter A; and, (

**b**) the existence of multiple solutions for the opposing flow case different parameter δ.

**Figure 2.**Variation of dimensionless velocity with Hartmann number for (

**a**) assisting flow and (

**b**) opposing flow.

**Figure 3.**Variation of dimensionless velocity with stretching velocity ratio when (

**a**) stretching in the flow direction and (

**b**) stretching in the opposite direction.

**Figure 4.**Variation of dimensionless temperature with Biot number in the presence of heat source for (

**a**) assisting flow and (

**b**) opposing flow.

**Figure 5.**Variation of dimensionless temperature with Biot number in the presence of heat sink for (

**a**) assisting flow and (

**b**) opposing flow.

**Figure 6.**Variation of dimensionless skin friction with several parameters in the presence of heat source for (

**a**) assisting flow and (

**b**) opposing flow.

**Figure 7.**Variation of dimensionless heat transfer rate with several parameters in the presence of heat source for (

**a**) assisting flow and (

**b**) opposing flow.

**Table 1.**Numerical results of Nusselt number—θ(0) for diverse Prandtl number when A = 1, A = 0, Ha = 0, and δ = 0.

Pr | Ramchandran et al. [26] | Hassanien and Gorla [30] | Lok et al. [27] | Ishak et al. [28] | Ali et al. [29] | Sharma et al. [3] | Present |
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1 | - | - | - | 0.8708 | 0.8708 | 0.8707 | 0.87078 |

10 | - | 1.9446 | - | 1.9446 | 1.9448 | 1.94463 | 1.94465 |

20 | 2.4576 | - | 2.4577 | 2.4576 | 2.4579 | 2.4576 | 2.4577 |

40 | 3.1011 | - | 3.1023 | 3.1011 | 3.1017 | 3.1011 | 3.1015 |

60 | 3.5514 | - | 3.556 | 3.5514 | 3.5524 | 3.55142 | 3.55148 |

80 | 3.9055 | - | 3.9195 | 3.9095 | 3.9108 | 3.90949 | 3.90919 |

100 | 4.2116 | 4.2337 | 4.2289 | 4.2116 | 4.2133 | 4.21163 | 4.21135 |

**Table 2.**Smallest eigenvalue

**γ**when λ = −0.2, Pr = 1, $A<0$ (for Shrinking surface) and $A>0$ (for Stretching surface).

$\mathit{\epsilon}$ | Ha | γ | - |
---|---|---|---|

1st Solution | 2nd Solution | ||

0.5 | 0.3 | 0.97533 | −0.09572 |

0.5 | 0.65753 | −0.06946 | |

−0.5 | 0 | 1.34857 | −0.75392 |

- | 0.5 | 1.02349 | −0.58327 |

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**MDPI and ACS Style**

Alarifi, I.M.; Abokhalil, A.G.; Osman, M.; Lund, L.A.; Ayed, M.B.; Belmabrouk, H.; Tlili, I.
MHD Flow and Heat Transfer over Vertical Stretching Sheet with Heat Sink or Source Effect. *Symmetry* **2019**, *11*, 297.
https://doi.org/10.3390/sym11030297

**AMA Style**

Alarifi IM, Abokhalil AG, Osman M, Lund LA, Ayed MB, Belmabrouk H, Tlili I.
MHD Flow and Heat Transfer over Vertical Stretching Sheet with Heat Sink or Source Effect. *Symmetry*. 2019; 11(3):297.
https://doi.org/10.3390/sym11030297

**Chicago/Turabian Style**

Alarifi, Ibrahim M., Ahmed G. Abokhalil, M. Osman, Liaquat Ali Lund, Mossaad Ben Ayed, Hafedh Belmabrouk, and Iskander Tlili.
2019. "MHD Flow and Heat Transfer over Vertical Stretching Sheet with Heat Sink or Source Effect" *Symmetry* 11, no. 3: 297.
https://doi.org/10.3390/sym11030297